# lfunc_search downloaded from the LMFDB on 14 April 2026.
# Search link: https://www.lmfdb.org/L/2/84/7.4
# Query "{'degree': 2, 'conductor': 84}" returned 6898 lfunc_searchs, sorted by root analytic conductor.

# Each entry in the following data list has the form:
#    [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin]
# For more details, see the definitions at the bottom of the file.



"2-84-21.11-c0-0-0"	0.20474731026602574	0.041921461061172206	2	84	"21.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.07324685680636028	0	2.56497434085969418732679777437	["ModularForm/GL2/Q/holomorphic/84/1/p/a/53/1", "ArtinRepresentation/2.84.6t5.b.a"]
"2-84-21.2-c0-0-0"	0.20474731026602574	0.041921461061172206	2	84	"21.2"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.07324685680636028	0	3.52084052442906496218604031148	["ModularForm/GL2/Q/holomorphic/84/1/p/a/65/1", "ArtinRepresentation/2.84.6t5.b.b"]
"2-84-1.1-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	2.24239796255929817450421188994	["EllipticCurve/Q/84/a", "ModularForm/GL2/Q/holomorphic/84/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/84/2/a/a"]
"2-84-1.1-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	2.51752502799515976298852414682	["EllipticCurve/Q/84/b", "ModularForm/GL2/Q/holomorphic/84/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/84/2/a/b"]
"2-84-12.11-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13632563922231528	0	1.22071947549038401216141071232	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/3"]
"2-84-12.11-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3482885685050119	0	1.23268990940303798440357755413	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/8"]
"2-84-12.11-c1-0-10"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3482885685050119	0	3.79170952090372095817767301368	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/7"]
"2-84-12.11-c1-0-11"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.21969514912354296	0	3.80992266774595779152159052255	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/9"]
"2-84-12.11-c1-0-2"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.18720966805897432	0	1.63003952872010285938024072312	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/2"]
"2-84-12.11-c1-0-3"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.18917136088669737	0	2.08163777518713281410239250840	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/6"]
"2-84-12.11-c1-0-4"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.21969514912354296	0	2.28496944310686873363744545772	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/10"]
"2-84-12.11-c1-0-5"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.009693912668569369	0	2.62369144497836808755406868877	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/11"]
"2-84-12.11-c1-0-6"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.18720966805897432	0	2.69342734417934637179760498892	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/1"]
"2-84-12.11-c1-0-7"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.009693912668569369	0	2.96303196562227100648239238537	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/12"]
"2-84-12.11-c1-0-8"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.13632563922231528	0	2.98550399227164041205278559022	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/4"]
"2-84-12.11-c1-0-9"	0.8189892410641031	0.6707433769787555	2	84	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.18917136088669737	0	3.00019171085947013424334186529	["ModularForm/GL2/Q/holomorphic/84/2/e/a/71/5"]
"2-84-21.17-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"21.17"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.15658019013969363	0	1.54619984643980010301992483740	["ModularForm/GL2/Q/holomorphic/84/2/k/a/17/1"]
"2-84-21.17-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"21.17"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.05043238263486531	0	2.41433171051343936092513682492	["ModularForm/GL2/Q/holomorphic/84/2/k/c/17/1"]
"2-84-21.17-c1-0-2"	0.8189892410641031	0.6707433769787555	2	84	"21.17"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17675314319363974	0	3.34339564043339402970541273188	["ModularForm/GL2/Q/holomorphic/84/2/k/b/17/1"]
"2-84-21.20-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"21.20"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13640723708574748	0	1.84955169590426068716106906195	["ModularForm/GL2/Q/holomorphic/84/2/f/a/41/2"]
"2-84-21.20-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"21.20"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.13640723708574748	0	3.01248503846707493834137147349	["ModularForm/GL2/Q/holomorphic/84/2/f/a/41/1"]
"2-84-21.5-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"21.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17675314319363974	0	1.66083755492376087956416999134	["ModularForm/GL2/Q/holomorphic/84/2/k/b/5/1"]
"2-84-21.5-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"21.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.15658019013969363	0	2.85316685493529774338497922753	["ModularForm/GL2/Q/holomorphic/84/2/k/a/5/1"]
"2-84-21.5-c1-0-2"	0.8189892410641031	0.6707433769787555	2	84	"21.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.05043238263486531	0	2.87999174247487663390578797746	["ModularForm/GL2/Q/holomorphic/84/2/k/c/5/1"]
"2-84-28.19-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"28.19"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.267959616430393	0	1.75982714799256931346701596506	["ModularForm/GL2/Q/holomorphic/84/2/o/b/19/2"]
"2-84-28.19-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"28.19"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.017576967474765205	0	1.77756495730142384033954453433	["ModularForm/GL2/Q/holomorphic/84/2/o/b/19/1"]
"2-84-28.19-c1-0-2"	0.8189892410641031	0.6707433769787555	2	84	"28.19"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.05786014617401908	0	1.95286796579453085706415655806	["ModularForm/GL2/Q/holomorphic/84/2/o/a/19/1"]
"2-84-28.19-c1-0-3"	0.8189892410641031	0.6707433769787555	2	84	"28.19"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.05257894226362498	0	2.73646517998044015543004009300	["ModularForm/GL2/Q/holomorphic/84/2/o/a/19/4"]
"2-84-28.19-c1-0-4"	0.8189892410641031	0.6707433769787555	2	84	"28.19"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.13977279743179902	0	2.96801562536090375430883642416	["ModularForm/GL2/Q/holomorphic/84/2/o/b/19/3"]
"2-84-28.19-c1-0-5"	0.8189892410641031	0.6707433769787555	2	84	"28.19"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.0007299271393613677	0	3.13236453059016307079577970554	["ModularForm/GL2/Q/holomorphic/84/2/o/b/19/4"]
"2-84-28.19-c1-0-6"	0.8189892410641031	0.6707433769787555	2	84	"28.19"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.21722447084147062	0	3.65590061496151109908035932221	["ModularForm/GL2/Q/holomorphic/84/2/o/a/19/3"]
"2-84-28.19-c1-0-7"	0.8189892410641031	0.6707433769787555	2	84	"28.19"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.4133875706501196	0	3.82782947239650031029925458891	["ModularForm/GL2/Q/holomorphic/84/2/o/a/19/2"]
"2-84-28.27-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"28.27"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.28890891597037877	0	0.985351524338355910375931631778	["ModularForm/GL2/Q/holomorphic/84/2/b/a/55/2"]
"2-84-28.27-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"28.27"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2264301710987845	0	1.82160855892319050731105368928	["ModularForm/GL2/Q/holomorphic/84/2/b/a/55/4"]
"2-84-28.27-c1-0-2"	0.8189892410641031	0.6707433769787555	2	84	"28.27"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.010135432237425775	0	1.85079309095007750368930363162	["ModularForm/GL2/Q/holomorphic/84/2/b/b/55/1"]
"2-84-28.27-c1-0-3"	0.8189892410641031	0.6707433769787555	2	84	"28.27"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.010135432237425775	0	2.38992235256116385514090455262	["ModularForm/GL2/Q/holomorphic/84/2/b/b/55/2"]
"2-84-28.27-c1-0-4"	0.8189892410641031	0.6707433769787555	2	84	"28.27"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.14591062505805094	0	2.59189804351623496803671798464	["ModularForm/GL2/Q/holomorphic/84/2/b/b/55/4"]
"2-84-28.27-c1-0-5"	0.8189892410641031	0.6707433769787555	2	84	"28.27"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.28890891597037877	0	2.74612645231323554418663994176	["ModularForm/GL2/Q/holomorphic/84/2/b/a/55/1"]
"2-84-28.27-c1-0-6"	0.8189892410641031	0.6707433769787555	2	84	"28.27"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.14591062505805094	0	3.32663223976659181890032032957	["ModularForm/GL2/Q/holomorphic/84/2/b/b/55/3"]
"2-84-28.27-c1-0-7"	0.8189892410641031	0.6707433769787555	2	84	"28.27"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.2264301710987845	0	3.46753522626254578920840203411	["ModularForm/GL2/Q/holomorphic/84/2/b/a/55/3"]
"2-84-28.3-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"28.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.4133875706501196	0	0.938493665518165874507254169903	["ModularForm/GL2/Q/holomorphic/84/2/o/a/31/2"]
"2-84-28.3-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"28.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.05786014617401908	0	1.93607887109344385434486206661	["ModularForm/GL2/Q/holomorphic/84/2/o/a/31/1"]
"2-84-28.3-c1-0-2"	0.8189892410641031	0.6707433769787555	2	84	"28.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.21722447084147062	0	2.04414996328470497698590750981	["ModularForm/GL2/Q/holomorphic/84/2/o/a/31/3"]
"2-84-28.3-c1-0-3"	0.8189892410641031	0.6707433769787555	2	84	"28.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.017576967474765205	0	2.32893792479702505689800989009	["ModularForm/GL2/Q/holomorphic/84/2/o/b/31/1"]
"2-84-28.3-c1-0-4"	0.8189892410641031	0.6707433769787555	2	84	"28.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13977279743179902	0	2.35976836196472391233673785529	["ModularForm/GL2/Q/holomorphic/84/2/o/b/31/3"]
"2-84-28.3-c1-0-5"	0.8189892410641031	0.6707433769787555	2	84	"28.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.05257894226362498	0	2.93046127186041610174628907038	["ModularForm/GL2/Q/holomorphic/84/2/o/a/31/4"]
"2-84-28.3-c1-0-6"	0.8189892410641031	0.6707433769787555	2	84	"28.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.0007299271393613677	0	3.12681587570989946224126052022	["ModularForm/GL2/Q/holomorphic/84/2/o/b/31/4"]
"2-84-28.3-c1-0-7"	0.8189892410641031	0.6707433769787555	2	84	"28.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.267959616430393	0	3.29935332341078946593648610885	["ModularForm/GL2/Q/holomorphic/84/2/o/b/31/2"]
"2-84-7.2-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.04034590610789225	0	2.31090490712168537397656940994	["ModularForm/GL2/Q/holomorphic/84/2/i/a/37/1"]
"2-84-7.4-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.04034590610789225	0	2.59165756938551738616972537442	["ModularForm/GL2/Q/holomorphic/84/2/i/a/25/1"]
"2-84-84.11-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3083839009665144	0	0.38989777422332225384693281936	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/4"]
"2-84-84.11-c1-0-1"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3511983361510115	0	1.41869927989089725511735896628	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/6"]
"2-84-84.11-c1-0-10"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1509079149397904	0	3.59072180711244618094878591963	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/12"]
"2-84-84.11-c1-0-11"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.24631889123359654	0	3.68218275236698978385342598332	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/7"]
"2-84-84.11-c1-0-2"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.020895697707647556	0	1.90342112112071006244386708090	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/1"]
"2-84-84.11-c1-0-3"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.16019294939587486	0	2.13605756298361284011251679922	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/11"]
"2-84-84.11-c1-0-4"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.08315859494905545	0	2.25840379201142614671954231218	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/5"]
"2-84-84.11-c1-0-5"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17210567300912263	0	2.31483020948501690108062067824	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/8"]
"2-84-84.11-c1-0-6"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.08095799004211175	0	2.35742743665918578297123624872	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/3"]
"2-84-84.11-c1-0-7"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.053119633008336324	0	2.75347416317491868892273338905	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/9"]
"2-84-84.11-c1-0-8"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.24228982321951714	0	2.83481319104630106689745666327	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/2"]
"2-84-84.11-c1-0-9"	0.8189892410641031	0.6707433769787555	2	84	"84.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.049054227190031045	0	2.99891680439949824354263468846	["ModularForm/GL2/Q/holomorphic/84/2/n/a/11/10"]
"2-84-84.23-c1-0-0"	0.8189892410641031	0.6707433769787555	2	84	"84.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.24228982321951714	0	1.44757443854930245061634394571	["ModularForm/GL2/Q/holomorphic/84/2/n/a/23/2"]
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"2-84-7.3-c20-0-6"	14.59285998994837	212.95156268623592	2	84	"7.3"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.1221205809213321	0	0.39605335712036362663231096600	["ModularForm/GL2/Q/holomorphic/84/21/m/a/73/6"]
"2-84-7.3-c20-0-7"	14.59285998994837	212.95156268623592	2	84	"7.3"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.48153918376136245	0	0.44070992817258670169644967402	["ModularForm/GL2/Q/holomorphic/84/21/m/b/73/5"]
"2-84-7.3-c20-0-8"	14.59285998994837	212.95156268623592	2	84	"7.3"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.17680713669108788	0	0.44073587139481724084236726159	["ModularForm/GL2/Q/holomorphic/84/21/m/a/73/1"]
"2-84-7.3-c20-0-9"	14.59285998994837	212.95156268623592	2	84	"7.3"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.29185674864926464	0	0.47877449535045080317316662789	["ModularForm/GL2/Q/holomorphic/84/21/m/b/73/14"]
"2-84-7.5-c20-0-0"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.3521305927947619	0	0.02542343353344711283394764845	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/5"]
"2-84-7.5-c20-0-1"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.1221205809213321	0	0.04867521572269988845305672224	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/6"]
"2-84-7.5-c20-0-10"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.29639557780246395	0	0.46660521403461636877060735043	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/12"]
"2-84-7.5-c20-0-11"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.16798103474708662	0	0.49705543424221625226536712334	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/4"]
"2-84-7.5-c20-0-12"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.11680344145126186	0	0.51604056030515203358048154677	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/6"]
"2-84-7.5-c20-0-13"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.21222609303285064	0	0.59728771423839688874555170270	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/2"]
"2-84-7.5-c20-0-14"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.17680713669108788	0	0.63591129064399702811865503608	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/1"]
"2-84-7.5-c20-0-15"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.0345624716409699	0	0.68926149290260826410361308476	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/12"]
"2-84-7.5-c20-0-16"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.061466889124856317	0	0.73867530934904313102864484860	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/9"]
"2-84-7.5-c20-0-17"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.026124942110695434	0	0.76221285200306225298987962974	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/13"]
"2-84-7.5-c20-0-18"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.21338245585687646	0	0.78363077767907137901280432966	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/13"]
"2-84-7.5-c20-0-19"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.22203138715021759	0	0.876214824643854091814740018702	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/8"]
"2-84-7.5-c20-0-2"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.3255418386769449	0	0.04928549439238770808745893915	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/10"]
"2-84-7.5-c20-0-20"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.08554793989031796	0	0.918571026374296238374275774605	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/9"]
"2-84-7.5-c20-0-21"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.32371524970260435	0	0.998186314793983600456740809175	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/8"]
"2-84-7.5-c20-0-22"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.4900579583823701	0	1.10418531298514732601304328399	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/7"]
"2-84-7.5-c20-0-23"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.33692388678160423	0	1.25002545197222850870161143939	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/11"]
"2-84-7.5-c20-0-24"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.40305368432204547	0	1.28966962393006883725248977350	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/10"]
"2-84-7.5-c20-0-25"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.29185674864926464	0	1.35779830284007239639065003448	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/14"]
"2-84-7.5-c20-0-26"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.48153918376136245	0	1.40196043589965070218609193459	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/5"]
"2-84-7.5-c20-0-3"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.3814988596864756	0	0.084060837993356049494212026287	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/7"]
"2-84-7.5-c20-0-4"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.10403152104739592	0	0.13865381007948790422679542124	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/3"]
"2-84-7.5-c20-0-5"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.35605732824561326	0	0.19612871172722636377709005352	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/1"]
"2-84-7.5-c20-0-6"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.1621821540167091	0	0.21354304409978995892505677002	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/4"]
"2-84-7.5-c20-0-7"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.35553973927150495	0	0.23899144485350435899902517870	["ModularForm/GL2/Q/holomorphic/84/21/m/b/61/11"]
"2-84-7.5-c20-0-8"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.4322967898706686	0	0.30488026914353079615145590410	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/3"]
"2-84-7.5-c20-0-9"	14.59285998994837	212.95156268623592	2	84	"7.5"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.07832797011055616	0	0.40712789208797837169027039584	["ModularForm/GL2/Q/holomorphic/84/21/m/a/61/2"]
"2-84-7.6-c20-0-0"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.4699614307020987	0	0.05564231598430769112235428622	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/24"]
"2-84-7.6-c20-0-1"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.09921391138730862	0	0.05699949557558500138692539762	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/5"]
"2-84-7.6-c20-0-10"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.1991688638933895	0	0.47114773514631855279749970517	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/17"]
"2-84-7.6-c20-0-11"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.009923278813394165	0	0.48530236011628061054993181597	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/15"]
"2-84-7.6-c20-0-12"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.2665872880386233	0	0.54183029609278218833062999788	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/23"]
"2-84-7.6-c20-0-13"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.09921391138730862	0	0.72934789038763716768984987769	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/22"]
"2-84-7.6-c20-0-14"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.020043688055067895	0	0.74417889541560338319806730645	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/26"]
"2-84-7.6-c20-0-15"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.13172703931291793	0	0.792628046104426220708358568195	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/7"]
"2-84-7.6-c20-0-16"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.1451520561376037	0	0.823663317361445269950410269109	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/21"]
"2-84-7.6-c20-0-17"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.2665872880386233	0	0.839137571320057199724648854940	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/4"]
"2-84-7.6-c20-0-18"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.231113049916232	0	0.871660245783995947922047070401	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/14"]
"2-84-7.6-c20-0-19"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.13172703931291793	0	0.901504438367080565828830254309	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/20"]
"2-84-7.6-c20-0-2"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.4496751654154556	0	0.099321879855447655537391663543	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/8"]
"2-84-7.6-c20-0-20"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.1991688638933895	0	0.934363079866931175026473974685	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/10"]
"2-84-7.6-c20-0-21"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.3661872434512183	0	1.04029646246437106937134933865	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/18"]
"2-84-7.6-c20-0-22"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.4699614307020987	0	1.10718674579164789384919819915	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/3"]
"2-84-7.6-c20-0-23"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.39355505183703987	0	1.18214111057849727125080616564	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/11"]
"2-84-7.6-c20-0-24"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.4090526837714269	0	1.32828213285136480035676666584	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/2"]
"2-84-7.6-c20-0-25"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.4496751654154556	0	1.43659671464012615301553829043	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/19"]
"2-84-7.6-c20-0-3"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.39355505183703987	0	0.13321009058549970402853531155	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/16"]
"2-84-7.6-c20-0-4"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.1451520561376037	0	0.13796195841000986945880954760	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/6"]
"2-84-7.6-c20-0-5"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.020043688055067895	0	0.24288411862260166965103672653	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/1"]
"2-84-7.6-c20-0-6"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.3661872434512183	0	0.24579721423748347092102947997	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/9"]
"2-84-7.6-c20-0-7"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	-0.231113049916232	0	0.43550595565495813941655079360	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/13"]
"2-84-7.6-c20-0-8"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.4090526837714269	0	0.44009817719926881120229931716	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/25"]
"2-84-7.6-c20-0-9"	14.59285998994837	212.95156268623592	2	84	"7.6"	[]	[[10.0, 0.0]]	20	true	true	false	false	0.009923278813394165	0	0.46739963466515189064238720921	["ModularForm/GL2/Q/holomorphic/84/21/d/a/13/12"]


# Label --
#    Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where

#     * $d$ is the degree of $L$.
#     * $N$ is the conductor of $L$.  When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions.
#     * q.k is the label of the primitive Dirichlet character from which the central character is induced.
#     * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation.
#     * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label.


#$\alpha$ (root_analytic_conductor) --
#    If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$.  It plays a role analogous to the root discriminant for number fields.


#$A$ (analytic_conductor) --
#    The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number
#    \[
#    A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N.
#    \]



#$d$ (degree) --
#    The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation

#    \[
#    \Lambda(s) := N^{s/2}
#    \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
#    \cdot L(s) = \varepsilon \overline{\Lambda}(1-s).
#    \]

#    The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor.



#$N$ (conductor) --
#    The **conductor** of an L-function is the integer $N$  occurring in its functional equation

#    \[
#    \Lambda(s) := N^{s/2}
#    \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
#    \cdot L(s) = \varepsilon \overline{\Lambda}(1-s).
#    \]


#    The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function
#     associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$.

#    In the literature, the word _level_ is sometimes used instead of _conductor_.


#$\chi$ (central_character) --
#    An L-function has an Euler product of the form
#    $L(s) = \prod_p L_p(p^{-s})^{-1}$
#    where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function.
#    Here, $N$ is the conductor of $L$.


#$\mu$ (mus) --
#    All known analytic L-functions have a **functional equation** that can be written in the form
#    \[
#    \Lambda(s) := N^{s/2}
#    \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
#    \cdot L(s) = \varepsilon \overline{\Lambda}(1-s),
#    \]
#    where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer
#    or half-integer,
#    \[
#    \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real},
#    \]
#    and $\varepsilon$ is the sign of the functional equation.
#    With those restrictions on the spectral parameters, the
#    data in the functional equation is specified uniquely.  The integer $d = J + 2 K$
#    is the degree of the L-function. The integer $N$ is  the conductor (or level)
#    of the L-function.  The pair $[J,K]$ is the signature of the L-function.  The parameters
#    in the functional equation can be used to make up the 4-tuple called the Selberg data.


#    The axioms of the Selberg class are less restrictive than
#    given above.

#    Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$.

#    For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$,
#    called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers.



#$\nu$ (nus) --
#    All known analytic L-functions have a **functional equation** that can be written in the form
#    \[
#    \Lambda(s) := N^{s/2}
#    \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
#    \cdot L(s) = \varepsilon \overline{\Lambda}(1-s),
#    \]
#    where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer
#    or half-integer,
#    \[
#    \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real},
#    \]
#    and $\varepsilon$ is the sign of the functional equation.
#    With those restrictions on the spectral parameters, the
#    data in the functional equation is specified uniquely.  The integer $d = J + 2 K$
#    is the degree of the L-function. The integer $N$ is  the conductor (or level)
#    of the L-function.  The pair $[J,K]$ is the signature of the L-function.  The parameters
#    in the functional equation can be used to make up the 4-tuple called the Selberg data.


#    The axioms of the Selberg class are less restrictive than
#    given above.

#    Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$.

#    For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$,
#    called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers.



#$w$ (motivic_weight) --
#    The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$.

#    If the L-function arises from a motive, then the weight of the motive has the
#    same parity as the motivic weight of the L-function, but the weight of the motive
#    could be larger.  This apparent discrepancy comes from the fact that a Tate twist
#    increases the weight of the motive.  This corresponds to the change of variables
#    $s \mapsto s + j$ in the L-function of the motive.


#prim (primitive) --
#    An L-function is <b>primitive</b> if it cannot be written as a product of nontrivial L-functions.  The "trivial L-function" is the constant function $1$.


#arith (algebraic) --
#    An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$  is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers.


#$\mathbb{Q}$ (rational) --
#    A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes
#    \[
#    L(s)=\prod_pL_p(p^{-s})^{-1}
#    \]
#    with $L_p\in \Z[T]$.


#self-dual (self_dual) --
#    An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real.


#$\operatorname{Arg}(\epsilon)$ (root_angle) --
#    The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$.


#$r$ (order_of_vanishing) --
#    The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point.

#    When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point).


#First zero (z1) --
#    The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$.

#    Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization).

#    The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number.


#Origin (instance_urls) --
#    L-functions arise from many different sources. Already in degree 2 we have examples of
#    L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions).

#    Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).